TABLE OF CONTENTS
ABSTRACT
CHAPTER ONE: INTRODUCTION
Page
1.0
Overview
1
1.1
Background of the Study
1
1.2
Statement of the Problem
2
1.3
Purpose of the study
2
1.4
Research Objectives
2
1.5
Research Questions
2
1.6
Research Hypothesis
2
1.7
Significant of the Study
3
1.8
Organisation of the Study
3
CHAPTER TWO: LITERATURE REVIEW
2.1
Conceptualization
4
2.2
Studies on Learning
5
2.3
Studies on Teaching
5
2.4
Approaches in Teaching
CHAPTER THREE: METHODOLOGY
3.1
Research Design
7
3.2
Population and Sample
7
3.3
Research Instrument
7
3.4
Validity and Reliability
8
3.5
Data collection and Intervention
8
3.6
Ethical Issues
8
3.7
Data Analysis
8
CHAPTER FOUR: RESULT AND DISCUSSION
4.1
Demographics
10
4.2
Research Question 1
11
4.3
Research Question 2
12
CHAPTER FIVE : CONCLUSION AND
REMMENDATIONS
5.1
Conclusion
13
5.2
Recommendation
13
ABSTRACT
One of the core topics covered in senior high school mathematics courses is linear inequality. To
evaluate students' performance in linear inequality, several research have been carried out.
However, it is rare to find linear inequality problems in the form of "ax+bdx+e" with "a,d0," as it
is shown in various research and textbooks used by Ghanaian students. This circumstance
prompts research questions about students' attempts to resolve a straightforward linear inequality
issue in this format. To do this, interviews were conducted after the written test, which was given
to 41 pupils from Form 2 at Mankessim Senior High School. The data also came from interviews
with teachers and student-used math textbooks. The data was then analyzed using the constant
comparative approach. The outcome demonstrates that most people answered the question using
algebraic techniques. It's interesting that most of them got it wrong. However, several of them
correctly applied algebraic operations. The others also completed expected-numbers solutions,
rewrote the question, verbalized the inequity, and provided blank answers. I also discovered that
nobody was aware of the existence of the all-numbers solution. It was discovered that this
situation is logically caused by how little the learning components care about how a method of
solving a linear inequality functions and potential solutions..
The purpose of This study aims to investigate the errors classes occurred by SHS2 students at
Mankessim Senior High Technical School, through analysis student responses to the items of the
study test, and to identify the varieties of the common errors and ratios of common errors that
occurred in solving inequalities. The researcher used an open-ended test with 10 items that
distributed information on two types of disparities in order to acquire the data (linear and
fractional). The descriptive analysis method was applied to the data analysis. The findings
indicated that some students had misunderstandings and misconceptions about how to solve
different forms of disparities...
CHAPTER ONE
INTRODUCTION
1.0. Overview
This chapter is about the Background to the Study, Statement of the Problem, Purpose of the
Study, Research Objective, Research Questions, Research Hypothesis, Significant of the
Study and Organization of the Study.
1.1. Background to the Problem
These days, mathematics becomes a nightmare for many students; also, mathematical
inequalities are considered an important mathematical topic as a prerequisite for many
subjects such as algebra, trigonometry and analytic geometry. Therefore, it falls to the
responsibility of educators to identify learning difficulties among students about the topics
that should be given to students in the light of these difficulties (Giltas and Tatar, 2011).
The inequality is a mathematical sentence built from expressions using one or more of the
symbols ( < > ≤ ≥) or to compare two quantities. Inequality solving means finding the
value(s) of variable that make the relationship correct order. So, inequality occupies an
important place in the basic math concepts, and being an important entry point for a lot of
mathematical topics such as equations and different kinds of functions (Salas, 1982, Ralph,
1997).
Therefore, the solution of equation (4 − 2𝑥 = 0) is the value that takes the variable (𝑥) and
makes the expression 4 − 2𝑥 is equal to zero, while solution of the inequality (4 − 2𝑥 < 0)
is all the values of (𝑥) that make the expression (4 − 2𝑥) a negative value.
solving equations and inequalities are considered to be an important topics in studying
properties and applications on functions, which require students to be aware and to
understand method of finding the solution set different types for each inequality and equation
( linear – non-linear and fractional). Kroll (1986) pointed that the mastering of solving
equations and inequalities affecting the students' improving performance in mathematics. The
equations and inequalities are two parts complement to each other, that don’t complete the
student knowledge in one part perfectly, but supplementing them.
The mental processes that are used in solving inequalities depending on the degree of
difficulty and type of inequality, where varies between the use of simple calculations to make
mathematical operations with difficult level. So, there are a lot of students with difficulties in
solving inequalities within the various stages of education. It can be due to the confusion
between the solution of equation and inequality, and sometimes students can't discriminate
between inequality solution procedures and equation, and some students don't take into
consideration what happen when inequality multiply by a negative number.
El-Shara' and Al-Abed (2010) See that the category of Common Mistakes When students can
be attributed to three sources represented with the nature of the subject, the student himself
and the teacher whose responsibility is to reduce the effect of each source of the school and
the student. Tsamir & Reshef (2006) Emphasized that the instruction approach used by
teachers may have an effect on the number of how the mistakes and their nature by their
students.
Recently, the interest in the identification of the common errors in the cognitive structure of
students increased before they learn mathematical concept. Also, several studies had
indicated that the mathematical knowledge is exist in the cognitive structure of the students,
and had considered being one of the most important factors affecting learning mathematics in
correct way (El-khateeb, 2015). The existence of the common misconceptions among
students could lead to a negative effect on the effectiveness of learning. This may be due to
the ignoring of the teachers' to the existence of perceptions and alternative interpretations of
learners before starting the new learning (shihap and Al-Jondey, 1999).
It has been observed during teaching “equations and inequalities" which is taught to the
Junior High School students as a compulsory requirement. There are some mistakes occurred
by some students when they solve different types of equations and inequalities: linear and
fractional.
Where it is noted that when the students' when multiply the inequality by a negative number
don't change direction of the inequality. Also, some students who don't exclude the zeros of
denominator from the solution set by solving the fractional inequality. So it must be taken
into consideration the importance of errors occurred by the students when teaching the topic
of solving linear equations and inequalities, in order to develop their skills and correct their
mistakes.
Abu- Guloah (2011) study aimed to identify the common errors at Numbers and Algebra for
the eighth graders' included in the international study test TIMSS 2007. The researcher used
the descriptive analytical method for the diagnosis of the most common errors of (369) male
and female students, including 193 male and 176 female students from primary eighth grade
who applied to diagnostic test. The researcher adopted (40%) and more as a ratio for the
existence of error. The study revealed the following findings: (21) of the skills emerged
within the previous experiences and the school book for the 8th grade encompass in TIMSS
2007. The percentage variation of the prevalent errors between the students in the diagnostic
test ranged between 13.5 % and 99.5 %. The skills group includes (15) Algebra and numbers
which consider the common error are 40% or more according to the researcher design.
The study of El-Shara and Al-Abed (2010) aimed to diagnose errors that occurred in solving
inequalities among mathematics majors at the University of Jordan. For the purpose of the
study, one test was developed and administered to 188 male and female students majoring in
mathematics who had completed Calculus 101.The results of the study revealed some
common errors, such as: misconceptions, confusing an inequality with an equation, using
commutative Multiplication in solving inequalities, and changing the direction of inequality
when multiplying by a negative number. Some other calculation errors and careless errors
were also recorded. The common errors ranged between 5.7% for changing the direction of
inequality when multiplying by a negative number, and 22.5% for conceptual errors. The
researchers recommended that faculty members should emphasize on the subject of
inequalities for fresh students and to administer tests in order to categorize them and develop
the appropriate treatment plans.
The study of Parish & Ludwig (1994) indicated its findings to the existence of errors at the
public high school and first year at the university students on the subject of algebra, including
the lack of writing equality symbol when solving equations, and their inability to find the
square root of the complete square terms or Algebraic Expressions. Students also have some
difficulties in using the language of mathematics.
A study Conducted by Bicer, etal (2014) aimed to determine whether pre-service teachers
have common difficulties and misconceptions about linear and quadratic inequalities. Two
tasks of inequalities openended were designed, and given to 57 participants. The study
showed that a number of pre-service teachers struggled with representing inequalities
solution in number line. They added or excluded values in their solutions by drawing a closed
circle on a number line instead of an open circle. Students also made basic arithmetic errors.
The most common errors were addition, subtraction, multiplication, division and the
distribution property. The results also indicate that not only the first year (pre-service
teacher) possesses difficulties and misconception with linear and quadratic inequalities, but
also second, third and fourth year pre-service teachers. The researchers due these
misconceptions might be transitional from teachers to their students.
Difficulties in Linear Inequality
The difficulties possibly come from previous learning or linear inequality learning itself. There
are several difficulties confronted by the students concerning linear equation and inequality as
follows:
i. A student makes mistake when carrying out addition, subtractions, multiplication, or
division of numbers.
ii. A student makes mistake when carrying out addition, subtractions, multiplication, or
division of algebraic expressions.
iii. A student does not follow the rules of order of arithmetical operations in numerical
expressions.
iv. A student misapplies a commutative property in calculating a division in numerical
expressions.
v. A student misuses a distributive property of a multiplication over an addition in algebraic
expressions.
vi. A student does not use an additive or multiplicative inverse in solving an equation.
vii. A student interprets a symbol has only a single rather than more than one value.
viii. A student substitutes a literal symbol in an equation with a particular value and the result
is incorrect.
ix.
A student experiences a conflict between the order in natural language and in algebraic
language.
x.
A student expects to have a numerical answer for an algebraic expression.
xi. A student adds or subtracts algebraic terms and numbers to get an algebraic term within
an algebraic expression.
xii. A student ignores local salience in an algebraic expression, such as the inequality sign
<, the variable 𝑥 , positive or a negative sign of an algebraic term.
xiii. A student ignores pattern salience in algebraic expression, such as an algebraic
expression with two terms and within a bracket.
xiv. A student does not understand the meaning of the equal sign as algebraic equivalence,
such as the student makes a notational error as a result of a combination of operations.
xv. A student does not understand the meaning of the equal sign as algebraic equivalence,
such as the student ignores the equal sign and applies an incorrect simplification on
algebraic expression.
xvi. A student mistranslates words or phrases into mathematical notations.
xvii. A student fails to formulate an equation or an inequality from the given word problem.
xviii. A student encounters a difficulty in interpreting a mathematical concept and pattern, in
substituting information into a formula and in using a formula.
xix. A student encounters difficulties in combining, in integrating, or in using information
either given in the task or given as a result of calculation in solving symbolic algebra
problems.
xx.
A student uses an arithmetical method to solve symbolic algebra problems.
xxi. A student encounters a difficulty in manipulating symbol when solving symbolic
algebra problems.
xxii. A student misapplies equation solving when simplifying algebraic expressions
1.2. Statement of the Problem
In light of the literature review and studies relevant to the solving inequalities, and common
mistakes which are located by the students, it is clear that some of these errors are common
among school students and university students. this underlines the importance of analyzing
errors classes and present them by providing feedback to Teachers of Mathematics, which
lead the researcher to study and diagnose the common errors, and classify them among the
students in Mankessim Senior High Technical School in the academic year 2021/2022.
1.3. Purpose of the Study
The purpose of This study aims to investigate the errors made by SHS2 students at
Mankessim Senior High Technical School, through analysis student responses to the items
of the study test, and to identify the varieties of the common errors and ratios of common
errors that occurred in solving inequalities. The importance of the current study played a
great role as result of the importance of solving inequalities in algebra, and in the
development of mathematical thinking among students, which in turn helps them to
continue their pre-university and university studies. It also highlights the importance of the
study through its attempt to analysis types of mistakes made by the students when they learn
the inequalities. In addition to enriching the studies conducted in this area and through the
identification of common errors in the solution of linear Inequalities, and find out its causes
in order to develop effective solutions for it.
1.4. Research Objective
The objective of This study is to investigate the errors made by SHS2 students at
Mankessim Senior High Technical School, through analysis student responses to the items
of the study test, and to identify the varieties of the common errors and ratios of common
errors that occurred in solving inequalities.
1.5. Research Questions
i.
What are the errors students makes when solving linear inequality questions?
ii.
What are the students’ typical answers on solving problem in linear inequalities?
iii.
What type of difficulties that students face when solving the problem?
iv.
What are the students’ explanations behind their answer?
v.
What are possible sources of these errors found from teachers and the textbooks
perspective?
1.6. Research Hypothesis
H0: Students understanding of linear inequalities will help them in their everyday lives.
H1: Students understanding of linear inequalities will not help them in their everyday lives.
1.7. Significant of the Study
The study will enable teachers to identify students’ errors and problems in solving linear
inequalities, modify their teachings and choose the appropriate methods of teaching for
students’ better understanding of the topic. As teachers find solutions to students’ difficulties
in solving linear inequalities, students will be able to understand it and apply it in their
everyday lives. It will also help external bodies like WAEC to make decisions about how to
set examination questions, the areas to set the questions from and how to set the questions.
As students understand and apply the concept of linear inequalities in their various homes,
they can use it to help their parents in their day to day activities, like buying and selling.
1.8. Organisation of the Study
The rest of the study involves;
Chapter Two which is about Literature Review
Chapter Three talks about Research Methodology
Chapter Four is about Results and Discussion and
Chapter Five is also about Conclusions and recommendations
CHAPTER TWO
LITERATURE REVIEW
2.1.Conceptualization
The signs for greater than (>) and less than (<) were introduced in 1631 in “Artis
Analyticae Praxis ad Aequationes Algebraicas Resolvendas.” The book was the work of
British mathematician, Thomas Harriot, and was published 10 years after his death in
1621. The symbols actually were invented by the book’s editor. Harriot initially used
triangular symbols which the editor altered to resemble the modern less/greater than
symbols. Interestingly, Harriot also used parallel lines to denote equality. However,
Harriot’s equal sign was vertical (II) rather than horizontal (=). The symbols for
less/greater than or equal to (< and >) with one line of an equal sign below them, were
first used in 1734 by French mathematician, Pierre Bouguer. John Wallis, a British
logician and mathematician, used similar symbols in 1670. Wallis used the greater
than/less than symbols with a single horizontal line above them.
As a discipline of study, inequalities do not have a long history. As a mathematical
concept, however, they were not foreign at all to ancient mathematicians (Bagni, 2005).
The ancients knew “the triangle inequality as a geometric fact” (Fink, 2000, p.120). They
were also aware of the arithmetic-geometric mean inequality, as well as the “isoperimetric
inequality in the plane” (Fink, 2000). Euclid used words ‘alike exceed’, ‘alike fall short’ or
‘alike are in excess of’ to compare magnitudes (Kline, 1972, p.69). The definition, “The
greater is a multiple of the less when it is measured by the less,” (Katz, 2009, p.74) shows
that the mathematicians of ancient times were adept at comparing magnitudes and
expressing the relationship between them.
Inequalities have been assisting mathematical discoveries from Classical Greek Geometry to
Modern Calculus and it took two millennia to change the status of inequalities from mere
support for some mathematics to Inequalities as a discipline of study (Fink, 2000). Today,
there are two journals of inequalities – The Journal of Inequalities and Applicationsand
The Journal of Inequalities in Pure and Applied Mathematics – as well as many other
mathematics publications that print papers with the “sole purpose to prove an inequality”
(Fink, 2000, p.118). The path that inequalities followed from Antiquity to the end of the
second millennium is investigated in the following sections.
Tanner (1962) indicates that, when producing the inequality signs, Harriot “took the equality
in Recorde's sign to reside not in the two lengths, but in the unvarying distance between
the two parallels” (p.166). According to Tanner (1962), Harriot modified the distance between
the two lines of the equal sign, to show that the biggest quantity lies on the side of the
biggest distance between the lines. Harriot used < to represent that the first quantity is less
than the second quantity and > to represent that the first quantity is greater than the second
quantity (Johnson, 1994) “The symbol for ‘greater than’ is > so that a > b will signify that a is
greater than b.
2.2. Studies on Learning Linear Inequalities
The study of Parish & Ludwig (1994) indicated its findings to the existence of errors at the
public high school and first year at the university students on the subject of algebra, including
the lack of writing equality symbol when solving equations, and their inability to find the
square root of the complete square terms or Algebraic Expressions. Students also have some
difficulties in using the language of mathematics. Several studies were conducted on this
inequality. In Ghana, it is clearly revealed that students faced difficulties when trying to solve
linear inequality tasks. Moreover, they failed to do operations involving algebraic form, such
as simplifying “4𝑥 + 7𝑥” to "11𝑥" This result shows that they treat it as same as dealing with
linear equation i.e. using algebraic manipulations. Although this method is acceptable, it can
lead to a misconception that is to solve an inequality means to solve appropriate equation. It
is not surprising since the problems in both of the inequality and the equation, apart from
equal symbol, are similar.
2.3.Studies on Teaching Linear Inequalities
The teacher Knowledge and understanding of the common error of students help to develop
strategies in teaching that address mathematical errors and misunderstandings, on the other
hand, the learner benefit from the error, and through verification of assumptions and
perceptions formed has started. In other words, we can say that mistakes can raise important
issues to discover more in mathematics, because teaching and learning mathematics is built
according to the sense of the importance of those errors that cannot be ignored or only
corrected.
In light of the above, and through the experience of the researcher, it is clear that knowledge
of the common mistakes occurred by students in teaching and learning mathematics is a
matter of concern, especially in the first stage of a Senior High School education. After
reviewing the educational literature and studies relevant to the inequalities topic, it is clear
that few studies have researched in the errors classification of students in solving the
inequalities in general at the Arab and local levels. At the local level, no studies have
addressed the solution of linear inequalities that include absolute value and diagnosis the
common errors, which occurred by the students at the level of school students or college
students. So this study was to bridge the gap as much as possible and to address with the
students in the Senior High Schools.
2.4.Approaches to teaching Linear Inequalities
A linear equation is a linear function that shows what one value is equal to. Similarly, a
linear inequality is also a linear function, but it shows a relationship between values using
“greater than”, “Less than”, “greater than or equal to”, or “less than or equal to” signs. Like
linear equations, you can teach a linear inequality by using algebra to isolate the variable.
Linear inequalities, however, have a few special rules that you need to pay close attention to.
There are a number of approaches you can teach linear inequalities. Some include; reversing
the inequality sign, maintaining the sign and others.
Teaching Linear Inequality
A. Maintaining the Sign
i.
Understand the inequality signs:
An inequality is like an equation, except instead of saying the two values are equal,
an inequality shows “greater than”, “Less than”, “greater than or equal to”, or “less
than or equal to” signs relationship. The sign < means “less than”, > means “greater
than”, ≤ means “less than or equal to” and ≥ means “greater than or equal to”
For instance,
2𝑥 +
3
> −15 + 𝑥
2
Means the value on the left side is greater than the value on the right side.
ii.
Combine like terms, or otherwise simplify the inequality
You can teach linear inequalities using the same algebraic approach you would use to
teach linear equation. You might need to combine variables, multiply to cancel out
fractions, or use other operations to make the numbers easier to work with.
Remember that you need to keep the inequality balanced, so whatever operation you
perform on one side of the inequality, you must also perform on the other side.
For instance, if teaching the inequality
2𝑥 +
3
> −15 + 𝑥
2
You would first multiply each term by 2 to get rid of the fraction
Ie.
3
2 (2𝑥 + ) > 2(−15 + 𝑥)
2
3
Expanding, you get 2(2𝑥) + 2 (2) > 2(−15) + 2(𝑥)
4𝑥 + 3 > −30 + 2𝑥
iii.
Move the variable to one side of the inequality (i.e. grouping of like terms)
To do this, add or subtract variables from one side of the inequality. Remember that
what ever you do to one side, you must also do to the other side.
For instance, in the inequality
4𝑥 + 3 > −30 + 2𝑥
to move the variable to one side, you subtract 2𝑥 from both sides of the inequality:
4𝑥 + 3 > −30 + 2𝑥
4𝑥 + 3 − 2𝑥 > −30 + 2𝑥 − 2𝑥
2𝑥 + 3 > −30
iv.
Isolate the variable:
In order to solve the inequality, the variable should be on one side, without
coefficients or constants. Divide to cancel out coefficients, and add or subtract to
remove constants. Once you have isolated the variable, you have solved the
inequality.
For instance
2𝑥 + 3 > −30
To isolate 𝑥, you need to subtract 3 from both sides, then divide both sides by 2
2𝑥 + 3 > −30
2𝑥 + 3 − 3 > −30 − 3
2𝑥 > −33
2𝑥
33
>−
2
2
𝑥 > −16
B. Reversing the Sign
1
2
i.
Approach the inequality as you would an equation
Use addition, subtraction, multiplication, and division to move the variable to
one side and isolate it.
ii.
Reverse the inequality sign when you multiply or divide by a negative
number
When using algebra to solve the inequality, pay close attention when you
multiply or divide. When you multiply or divide the inequality by a negative
number, the direction of the inequality sign must be reversed.
For example, to solve the inequality −5𝑥 > 20, you need to divide each term
by −5 to isolate the variable. Thus, you need to reverse the direction of the
inequality sign.
𝑡ℎ𝑎𝑡 𝑖𝑠
−5𝑥 > 20
−
5𝑥
20
>
−5 −5
𝑥<4
iii.
Reverse the inequality sign whenever you take the reciprocal of both sides
This is only the case if both sides are negative or are positive. The reciprocal
1
of 𝑥 is 𝑥
1
For instance, to solve the inequality 6 < 𝑥, you would isolate the 𝑥 by taking
the reciprocal of both sides. Since both sides are positive, you need to reverse
the inequality sign.
6<
1
𝑥
1
>𝑥
6
1
In Ghanaian teaching and learning, if you want to solve 6 < 𝑥, you can say
let’s “cross multiply”. That is
6<
1
𝑥
𝑏𝑦 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑖𝑛𝑔, 𝑦𝑜𝑢 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 𝑥
⇒6×𝑥 <
1
×𝑥
𝑥
6𝑥 < 1
𝑑𝑖𝑣𝑖𝑑𝑒 𝑒𝑎𝑐ℎ 𝑡𝑒𝑟𝑚 𝑏𝑦 6
𝑥<
1
𝑜𝑟
6
1
>𝑥
6
In general, if you want to choose an operation that makes the inequality simpler. If an
unknown appears on both sides of the inequality, add or subtract terms to get all of the
variables on the same side so that you can combine like terms on that side. Reducing the
number of "𝑥" appears in your inequality makes it simpler. If the unknown only appears
once, but has a coefficient in front of it, divide both sides by the coefficient. A plain 𝑥 in
an inequality is simpler than a 7𝑥.
CHAPTER THREE
METHODOLOGY
3.1. Research Design
The study used a pretest and posttest. The purpose of a teaching experiment could be one of
curriculum testing and development or one of theory construction. This teaching experiment
was designed to assess form two students in Senior High Schools to learn to solve linear
inequality. In this teaching experiment, lessons were planned and taught to the students. It
covered two contact periods of a total duration of two hours.
3.2. Population and Sample
My study was taken at Mankessim Senior High School in the Mfantsiman district. The
targeted people were form two students at the General Arts department. There were 41
students in total of which 22 were males and 19 females in the study. Their ages ranges
from 14 to 18 years.
3.3. Research Instruments
The study tool consisted a test of solving linear inequality, which was built in light of the
expected appearance in student responses errors through types of linear inequalities. The
test included 10 multiple-choice test and (5) essay items (open- ended question), three
items for each type of Inequality.
PRE-TEST
10 multiple–choice test items were constructed from linear inequality topic in the Second
Term Scheme of Work. All the common topics among the students constitute pre–requisite
skills for the solving of linear inequalities. These items were drawn from areas involving
fraction, maintaining the sign, reversing the sign and others. These are samples of the
pretest items;
Solve the following inequalities (show workings in each question)
1. 3 − 2𝑥 > 5
a. 𝑥 < 1
b. 𝑥 > −1
c. 𝑥 < −1
d. 𝑥 = −1
2.
1
𝑥
<2
1
a. 𝑥 < 2
1
b. 𝑥 > 2
1
c. 𝑥 > − 2
1
d. 𝑥 < − 2
3. 𝑥 + 2 < 4
a. 𝑥 < 6
b. 𝑥 < −2
c. 𝑥 > 2
d. 𝑥 < 2
4. −2𝑥 ≤ 10 + 3𝑥
a. 𝑥 ≥ −5
b. 𝑥 ≥ 2
c. 𝑥 > 5
d. 𝑥 < 5
5. 3𝑥 − 13 > 26
a. {𝑥: 𝑥 > 13}
b. {𝑥: 𝑥 < 13}
c. {𝑥: 𝑥 ≥ 13}
d. {𝑥: 𝑥 ≤ 13}
6.
𝑥
−2
+ 4 > 17
a. 𝑥 < −26
b. 𝑥 > −26
c. 𝑥 > −
d. 𝑥 < −
13
2
13
2
7. 14 > 8 − 2𝑥
a. 𝑥 < 6
b. 𝑥 > 3
c. 𝑥 < −3
d. 𝑥 > −3
8. 13𝑥 − 15 ≥ 33 + 25𝑥
a. 𝑥 ≤ −14
b. 𝑥 > 14
c. 𝑥 ≤ −4
d. 𝑥 ≤ 4
9. 4 − 2𝑥 ≤ (𝑥 + 1)
1
a. 𝑥 ≤ 5
b. 𝑥 ≤ 3
1
c. 𝑥 ≥ 5
d. 𝑥 ≥ 1
3
1
10. 2 (1 − 𝑥) > 4 − 𝑥
a. 𝑥 < 5
5
b. 𝑥 < 2
5
c. 𝑥 > 2
d. 𝑥 > 5
POSTTEST
The posttest constituted five (5) essay type items based on linear inequality. The items required
students to find the value(s) of x in the following linear inequalities.
1. Solve 4(𝑥 + 2) − 1 > 5 − 7(4 − 𝑥)
2. Twice a number is subtracted from four and the result is less than 10. Given that 𝑥 is the
number, find the value of 𝑥.
3. Solve the inequality 𝑥 + 7 < 20 and represent your answer on a number line.
1
1
1
4. Solve 3 𝑥 − 4 (𝑥 + 2) ≥ 3𝑥 − 1 3
3
1
17
5. Determine the solution set of 4 𝑥 + 2 ≤ 16
SCORING
A marking scheme was drawn for both the pre-test and post-tests, such that there was no
differential on the basis of the treatment so that no one treatment gained an undue
advantage over the other. Each item of the pretest was awarded 3 marks and that of the
posttest was 5 marks. Every logical Mathematical step was awarded a mark including the
final answer.
3.4. Reliability and Validity
To check the validity, the test was presented to a group of permanent teachers in my
department including my mentor and the Head Of Department. Each of them was given the
test items and a list of common errors that have been prepared, and were asked to express
their opinions about the item’s fitness and suitability of the target group of the Mankessim
Senior High School. After reviewing the opinions of the teachers and suggestions have been
modified, to achieve the purpose of the research and investigated the errors classes, the
application of tests procedures and instructions require that the student shows the steps
resolved in detail, and in which is standing on the strengths and weaknesses in student
performance, so the tests in this way can be considered that achieved a standard of validity.
The reliability compute by using pre-test and post-test, by applying the tests on an
exploratory sample consisted of (41) students, with an interval of (two) weeks, who
completed studying the equation and inequalities in the core mathematics. The Pearson
correlation coefficient was (0.89) between the average performance of students in the first
time and repetition (Oadeh, 2005; Gronlaund, 1990).
3.5. Data collection and Intervention
Qualitative methods for data collection included the Initial Assessment Questions (Mack,
1985), student work samples, and videotaped and audiotaped sessions of pupils working in
groups. All data were collected and care was taken to prevent data collection from
interfering with classroom activities.
3.6 Ethical Issues
Before conducting my study, I obtained the approval of my institutional review board, my
Head of Department, research ethics committee and the students.
3.7. Data Analysis
My study was designed to investigate the errors students make when solving linear
inequalities. I considered it necessary that my findings be accurate and trustworthy.
“Teachers who engage in action research can take steps to ensure that their data are
trustworthy through triangulation” (Feldman & Minstrell, 2000, p.437). Triangulation of
findings on students’ understanding and performance regarding inequality concepts,
operations and procedures was provided through several different overlapping data sources
and methods including: Pre and post assessments, and student work samples. The pre and
post assessments and student work samples were read and reread in order to note persistent
themes or common threads (Gay & Airason, 2003, p.231).
CHAPTER FOUR
RESULTS AND DISCUSSION
4.1 Demographics
Age * Gender Crosstabulation
Count
Table 1 shows the crosstab of gender of
Gender
Male
students by their age
Female
2
4
6
From the table, a total of 41 students were
15-17
18
12
30
sampled, out of this, 24 were males and 17
18-20
4
1
5
24
17
41
below 15
Age
Total
Total
were females. There is a total of 6 students
below 15 years, out of this, 2 are males and 4
are females. 30 students are between the ages of 15 - 17, of which 18 are males and 12 are
females. Also, there is a total 5 students between the ages of 18 – 20, of which 4 are males and 1
are females.
Score * Gender Crosstabulation
Count
Gender
Pre- test
Score
Total
Male
Total
Table 2 shows a crosstab of gender
Female
less than 10
6
7
13
10 - 14
6
8
14
15 - 19
1
3
4
20 - 24
3
4
7
25 - 30
3
0
3
19
22
41
against the score.
From the table, a total of 41 students
participated in the pretest. Out of this, 13
scored less than 10 marks, of which 6 are
male and 7 are female. 14 scored between 10 – 14, of which 6 are male and 8 are female. 4
scored between 15 – 19, of which 1 of them is a male and 3 are female. Between 20 -24, there
were 7 students, of which 3 were males and 4 females. Lastly, 3 scored between 25 – 30, of
which all of them are males.
Report
Score
Mean
2.34
Std. Deviation
1.296
Variance
1.680
% of Total Sum
100.0%